1.2 Gauge fields as Kaluza–Klein vectors

Since the work of Kaluza and Klein, one can conceive that our electromagnetic gauge field could
originate from a Kaluza–Klein vector of a higher-dimensional spacetime of the form , where
is our spacetime, is compact and contains at least a cycle (the total manifold might not
necessarily be a direct product). Experimental constraints on such scenarios can be set from bounds on the
deviation of Newton’s law at small scales [197, 2].

If our electromagnetic gauge field can be understood as a Kaluza–Klein vector, it turns out that
it is possible to account for the entropy of the Reissner–Nordström black hole in essentially the same way
as for the Kerr black hole [159]. This mainly follows from the fact that the electric charge becomes
an angular momentum in the higher-dimensional spacetime, which is on the same
footing as the four-dimensional angular momentum lifted in the higher-dimensional
spacetime.

Assumption

We will assume throughout this review that the electromagnetic gauge field
can be promoted as a Kaluza–Klein vector.

As far as the logic goes, this assumption will not be required for any reasoning in Section 2, even though
it will help to understand striking similarities between the effects of rotation and electric charge. The
assumption will be a crucial input in order to formulate the Reissner–Nordström/CFT correspondence and
its generalizations in Section 4 and further on. This assumption is not required for the Kerr/CFT
correspondence and its (extremal or non-extremal) extensions, which are exclusively based on the
axial symmetry of spinning black holes.

In order to make this idea more precise, it is important to study simple embeddings of the gauge
field in higher-dimensional spacetimes as toy models for a realistic embedding. In asymptotically-flat
spacetimes, let us introduce a fifth compact dimension , where is the length of the
Kaluza–Klein circle and let us define

The metric (2) does not obey five-dimensional Einstein’s equations unless the metric is complemented by
matter fields. One simple choice consists of adding a gauge field , whose field strength is
defined as

where is the four-dimensional Hodge dual. The five-dimensional metric and gauge field are then
solutions to the five-dimensional Einstein–Maxwell–Chern–Simons theory, as reviewed, e.g.,
in [185].

These considerations can also be applied to black holes in anti-de Sitter spacetimes. However, the
situation is more intricate because no consistent Kaluza–Klein reduction from five dimensions can give rise
to the four-dimensional Einstein–Maxwell theory with cosmological constant [204]. As a consequence, the
four-dimensional Kerr–Newman–AdS black hole cannot be lifted to a solution of any five-dimensional
theory. Rather, embeddings in eleven-dimensional supergravity exist, which are obtained by adding a
compact seven-sphere [69, 109].

Therefore, in order to review the arguments for the Reissner–Nordström/CFT correspondence and its
generalizations, it is necessary to discuss five-dimensional gravity coupled to matter fields. We
will limit our arguments to the action (1) possibly supplemented by the Chern–Simons terms

where are constants. This theory will suffice to discuss in detail the embedding
(2) – (3) since the five-dimensional Einstein–Maxwell–Chern–Simons theory falls into that class
of theories. We will not discuss the supergravities required to embed AdS–Einstein–Maxwell
theory.

Let us finally emphasize that even though the scale of the Kaluza–Klein direction is arbitrary as
far as it allows one to perform the uplift (2), it is constrained by matter field couplings. For example,
let us consider the toy model of a probe charged massive scalar field of charge in
four dimensions, which is minimally coupled to the gauge field. The wave equation reads as

where the derivative is defined as . This wave equation is reproduced from a
five-dimensional scalar field probing the five-dimensional metric (2), if one takes

and if the five-dimensional mass is equal to . However, the five-dimensional scalar is
multivalued on the circle unless

This toy model illustrates that the scale can be constrained from consistent couplings with matter. We
will use this quantization condition in Section 6.4.